Enough average capacity, yet a queue
Why can a queue form even when average demand equals capacity?
Start with a concrete question
Original synthetic service desk: eight one-minute intervals, arrivals enter before service, and unfinished people remain in a first-in queue.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
What do you expect?
Predict whether six arrivals per minute and six service places will queue when arrivals come in alternating bursts.
Change one thing. Inspect what happens.
- Start with demand and capacity both 6 and steady arrivals; inspect all eight updates.
- Select bursts without changing average demand. Count how often the queue is nonzero.
- Raise capacity to 8. Explain why the burst queue shrinks but does not vanish.
- Compare improving average capacity with smoothing arrivals; write one operational limitation for each.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Minute 1: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 2: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 3: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 4: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 5: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 6: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 7: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
- Minute 8: previous queue 0 + arrivals 6 - served 6 = 0 waiting.
Equal average demand and capacity can still create temporary queues when arrivals are bunched. End-minute queue totals are a discrete congestion measure, not exact individual waiting times.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
What the experiment explains
Original synthetic service desk: eight one-minute intervals, arrivals enter before service, and unfinished people remain in a first-in queue.
A queue is unfinished demand carried into the next interval. Service cannot be saved from an idle minute for later use.
With bursts, minute one receives 12 and serves 6, leaving 6. Minute two receives none and serves those 6. Average arrivals still equal six, but some people wait.
The total of end-minute queues is a discrete congestion indicator. Because arrivals and service occur within simplified intervals, it is not an exact average waiting time.
Read the relationship
Next queue = max(0, previous queue + arrivals − capacity). Served = min(previous queue + arrivals, capacity); all counts are people per interval or people waiting.
Use the trace as evidence
The trace recomputes from your current controls. Change one control at a time, record what changed, and compare the result with your prediction. The bars use the same quantities as the numerical trace; a larger bar is not automatically a better outcome.
Boundaries of the model
- Fixed interval service, no abandonment or priority, identical service needs, and deterministic arrivals.
- Eight minutes is a bounded scenario, not proof of long-run stability under random demand.
Continue with When faster work creates a bigger pile to investigate a different mechanism.
Try a fresh case
New case: begin empty, capacity 5 each minute, arrivals 8, 0, 8, 0 over four minutes. What is the sum of the four end-minute queue counts?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Draw arrivals and service as tokens and explain why idle capacity cannot be stored.
College investigation
Implement individual arrival timestamps and compare exact waiting time with this interval approximation.
Workplace investigation
Draft a staffing experiment that records demand variation, service times, and an acceptable waiting-time target.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.